ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Burgers centered rarefaction and trace

Example

Assume the Axiom of Countable Choice. Let $T > 0$ , let $a, b \in R$ with $a < b$ , and set $F (z) = z^{2} / 2$ . For $t > 0$ define $u (t, x) = {\begin{cases} a, & x \leq a t, \\ x / t, & a t < x < b t, \\ b, & x \geq b t . \end{cases}$ Let $u_{0} (x) = a$ for $x < 0$ and $u_{0} (x) = b$ for $x > 0$ , with any value at $x = 0$ . Then $u$ is continuous for $t > 0$ , satisfies the initial-value weak identity for $u_{t} + \partial_{x} F (u) = 0$ , and satisfies with equality the initial entropy identity for every convex $η \in C^{2} (R)$ with compatible flux $q^{'} = η^{'} F^{'}$ and every Kružkov pair $η_{k} (z) = ∣ z - k ∣$ , $q_{k} (z) = sgn (z - k) (F (z) - F (k))$ , where $sgn (0) = 0$ . For every compact interval $K \subset R$ and $t > 0$ , $\int_{K} ∣ u (t, x) - u_{0} (x) ∣ d x \leq (b - a) (∣ a ∣ + ∣ b ∣) t,$ so $u (t, \cdot) \to u_{0}$ strongly in $L_{l o c}^{1} (R)$ as $t ↓ 0$ .

Example references

strictly convex scalar Riemann construction

published Burgers shock and rarefaction theorem

initial weak and entropy formulations

Axiom of Countable Choice

Facts & Assumptions

Given: Countable Choice; the states $a < b$ ; the Burgers flux, fan, initial data, and entropy pairs in the Example.

[F1]

Strictly convex scalar Riemann construction: for $F \in C^{2} (R)$ with $F^{''} > 0$ on $[a, b]$ , the inverse-characteristic fan in the Statement satisfies the initial-value weak identity, compatible convex entropy inequalities, and Kružkov inequalities.

[F2]

Riemann shock and rarefaction waves for $u_{t} + (u^{2} / 2)_{x} = 0$ are weak solutions, and the shock is an entropy solution: assuming Countable Choice, the Burgers fan is a weak solution and satisfies every declared convex $C^{1}$ and Kružkov entropy inequality, with equality.

[F3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the choice assumption used by the imported Lebesgue weak formulation.

[F4]

Distributional systems and entropy flux pairs: for smooth tests compactly supported in $[0, T) \times R$ , the scalar initial weak identity is $\int_{0}^{T} ⁣ \int_{R} (u φ_{t} + F (u) φ_{x}) d x d t + \int_{R} u_{0} φ (0, x) d x = 0$ ; the entropy inequality replaces $(u, F, u_{0})$ by $(η, q, η (u_{0}))$ , requires a nonnegative test, and requires the whole integral to be nonnegative.

[A1]

$a < b$ .

Proof

1.1givenF1F2F3F4A1

The Burgers flux satisfies $F^{'} (z) = z$ and $F^{''} (z) = 1 > 0$ on $[a, b]$ . Thus [F1] applies, and its inverse profile is $u = x / t$ for $a t < x < b t$ , with the two constant branches displayed in the Example. It gives the weak and entropy identities with the specified initial trace in the formulation of [F4]; [F2] independently confirms the same Lebesgue formulation under [F3].

2.1givenF1A1algebra∎

In the fan, $u_{t} = - x / t^{2}$ and $u_{x} = 1 / t$ , so $u_{t} + u u_{x} = 0$ ; the constant branches solve the equation, and the traces match at $x = a t$ and $x = b t$ . The fan values stay in $[a, b]$ . The profile and initial step agree outside $[\min (0, a t), \max (0, b t)]$ , whose length is at most $t (∣ a ∣ + ∣ b ∣)$ , and their values differ by at most $b - a$ . This proves the stated local $L^{1}$ estimate, including when both characteristic speeds have the same sign.

Burgers centered rarefaction and trace

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources